Infinite Sequences with Finite Cross-Correlation-II

Abstract

This extends a study in [1], from SETA-2010. We consider two infinite sequences of positive integers, $A= \{a_k\} ^\infty_{k=1}$ and $B = \{b_k\}_{k=1}^\infty$, and the related sequences $\bar{A} = \{\alpha_j\}_{j=1}^\infty$ and $\bar{B} = \{\beta_j\}_{j=1}^\infty$, where α _j  = 1 if j ∈ A, α _j  = 0 if j ∉ A, and β _j  = 1 if j ∈ B, β _j  = 0 if j ∉ B. We call C _AB (τ) the cross-correlation of A and B where C _AB (τ) is the un-normalized infinite-domain cross-correlation of $\bar{A}$ and $\bar{B}$, i.e.

$$ C_{AB}(\tau) = \sum \limits_{i=1}^\infty \alpha_i\beta_{i+\tau},\: for all\: \tau \in Z. $$

((1))

Our interest is confined to sequence pairs A,B for which C _AB (τ) is finite for all τ ∈ Z. Our interest is greater if C _AB (τ) < K for some uniform bound K, for all τ ∈ Z, and especially great if K = 1. We are specifically interested in possible lower bounds on the rates of growth of A and B for a given value of K. This paper provides extensive data for the case K = 1.